Interactive guide
68, 95, 99.7: the same three numbers, every normal curve
For any normal distribution, no matter its mean or spread, about 68% of the data sits within one standard deviation of the mean, 95% within two, and 99.7% within three. Those aren't rounded guesses, they're exact areas under the curve, the same three numbers whether σ is 1 or 100. Drag the mean and spread below and watch the bands stay put in standard deviations even as the curve itself moves and stretches.
A property of shape, not of scale
Measure any distance in standard deviations instead of raw units, and a normal curve always looks identical: the same bell, the same proportions, every time. That's exactly why the 68/95/99.7 numbers never change with μ or σ, they're areas under the one true, unchanging shape, just relabeled in whatever units the real data happens to use.
The exact figures are 68.27%, 95.45%, and 99.73%, the area under the standard normal curve between −1 and 1, between −2 and 2, and between −3 and 3. "68-95-99.7" is just the rounded version people actually memorize.
This only holds because the shape is normal. A skewed or bimodal population, like the ones on the Central Limit Theorem page, won't put 68% of its data within one standard deviation at all, until you're looking at the sampling distribution of a large-enough sample mean instead of the raw data itself.
Reading the diagram
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1
Three nested bands, one curve
Darkest at the center (±1σ), lightest at the edges (±3σ).
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2
Drag the value marker
Its z-score and percentile update live, computed from the real normal CDF, not looked up from a table.
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3
Generate real samples
Actual random draws from this exact μ, σ, counted into each band and compared to the theoretical share.
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Watch small samples wobble, large ones settle
The same Law of Large Numbers idea: more draws, closer to the theoretical 68/95/99.7.
Drag the marker under the curve.
±1σ (68.27%) ±2σ (95.45%) ±3σ (99.73%)
- Marker value
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- z-score
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- Percentile
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Distribution
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Exact vs. rounded rule
| Band | Exact | Rounded rule |
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Check it with real samples
| Band | Theoretical | Empirical |
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Draw samples to compare theory against real random data.
Draw only 100 samples and the empirical shares will usually be close but visibly off, 66% here, 71% there, nothing stays exactly 68.27% at that scale. Draw 1,000 and they tighten up considerably. The theoretical 68/95/99.7 numbers describe the exact population; any finite sample is a noisy estimate of it, the same sample-size-versus-precision tradeoff behind the Confidence Intervals page.
The jargons
All of these describe distance from the mean, just measured in different units.
For any normal distribution, roughly 68%, 95%, and 99.7% of the data falls within 1, 2, and 3 standard deviations of the mean.
(x−μ)/σ: how many standard deviations a value sits from the mean, positive above, negative below.
The share of the distribution lying at or below a given value, exactly the normal CDF evaluated there.
The one normal curve with μ=0 and σ=1. Every other normal curve is this same shape, just shifted and rescaled.
Often flagged past ±2 or ±3 standard deviations, precisely because the empirical rule says so little of the data should be out there.
The empirical rule needs a normal shape. Chebyshev's inequality gives weaker but universal bounds (at least 75% within 2 SD) for any distribution at all.